Question

How potential energy is equal to -dU/dx ?

Answer 1

They are not equal.

Explanation:

It is the force that is equal to `-(dU)/dx`

`F_x = - (dU)/dx`

where F_x is the force in x-diretion, and U is the potential energy. dU/dx is how potential energy in x-direction.

Example 1, for a spring system
`U = 1/2kx^2`

`rArr F_x = - (dU)/dx =- kx`

Obviously, `F_x` is the restoring force of the spring when it is compressed or stretched, it's direction of which is always opposite to the compression or extension.

Example 2, for gravity

`U= -G(m_1m_2)/r`

`rArr F_r = -(dU)/(dr) =G(m_1m_2)/r^2`

Hence `F_r` is the restoring force tugging on a mass body, like an invisible spring.

A restoring force that are independent of pathways is also called a conservative force.

Finally, the U can come in many forms, many of them are energy stored in distorted shapes such a bow, a twisted or bent rod or from bodies that reside in force fields such a magnets (or magnetic dipoles) in magnetic fields, a charge in electric field etc etc.